Filipa Nunes Nogueira
Contributions to positive systems
control: mathematical methods and
applications
University of Porto
December, 2013
Contributions to positive systems
control: mathematical methods and
applications
Thesis submitted to the University of Porto, Portugal, in fulfillment of the requirements for the degree of Doctor of Philosophy in Applied Mathematics, under the supervision of
Professor Maria Paula Macedo Rocha Malonek, Full Professor at the Faculty of Engineering of the University of Porto, and Professor Teresa Maria de Gouveia Torres
Feio Mendon¸ca, Assistant Professor at Faculty of Science of the University of Porto.
University of Porto December, 2013
Acknowledgments
In this space I would leave my sincere thanks to all those who somehow contributed to this work became possible.
First of all, I would like to express a special gratitude to my supervisors, Professor Paula Rocha Malonek and Professor Teresa Mendon¸ca, for their guidance, support, encouragement, availability, dedication, patience, and friendship throughout this re-search. Without their expertise, this work would not be possible.
I wish to express my thankfulness to dr. Rui Rabi¸co (anesthesiologist at Unidade Local de Sa´uude de Matosinhos - Hospital Pedro Hispano) and to dr. Sim˜ao Esteves (anesthesiologist at Hospital Santo Ant´onio - Porto) for the possibility that they provided me to follow closely all the work related to general anesthesia and the chance to perform the work developed in my thesis.
I am thankful for the financial support given by FCT - Portugal through the grant SFRH/BD/48314/2008. I also acknowledge GALENO - Modeling and Control for per-sonalized drug administration, Funding Agency: FCT, Programme: FCT PTDC/SAU-BEB/103667/2008, and the support of FEDER founds through COMPETE–Operational Programme Factors of Competitiveness (“Programa Operacional Factores de Com-petitividade”) and by Portuguese funds through the Center for Research and Devel-opment in Mathematics and Applications (University of Aveiro) and the Portuguese Foundation for Science and Technology (“FCT–Funda¸c˜ao para a Ciˆencia e a Tecnolo-gia”), within project PEst-C/MAT/UI4106/2011 with COMPETE number FCOMP-01-0124-FEDER-022690.
I thanks to all my laboratory colleagues Am´elia, Haider, Igor, Lu´ıs, M´ario, Sofia, specially to Ana Filipa, Juliana and Tiago, for their fellowship, support and for the entertainment they provided me.
Last but not least, one special thanks to my parents, brother and Pedro who always gave me forces in difficult times and supported me from the first time.
Resumo
Neste trabalho ´e apresentada uma lei de controlo positiva para sistemas positivos do tipo multi-input, que garante o seguimento assint´otico de um valor de referˆencia desejado. Esta lei de controlo pode ser vista como uma generaliza¸c˜ao de uma outra proposta na literatura para o controlo da massa total em sistemas SISO comparti-mentais, no entanto ´e adequada para uma classe mais ampla de sistemas positivos. O controlador aqui proposto ´e aplicado no controlo da profundidade de anestesia (DoA), por meio da administra¸c˜ao de propofol e remifentanil, quando se utiliza um modelo de Wiener de parˆametros parcimoniosos (PPM), recentemente introduzido na literatura. O seu desempenho ´e ilustrado por meio de simula¸c˜oes realistas. Tamb´em ´e apresentado um novo procedimento de estima¸c˜ao para os parˆametros do PPM. Uma vez que este modelo ´e parametrizado com parcim´onia, ´e poss´ıvel aplicar um procedimento simples de estima¸c˜ao de parˆametros, com base na resposta ao degrau do paciente. Mostra-se que as estimativas dos parˆametros obtidas desta forma proporcionam bons resultados quando utilizadas para desenhar a lei de controlo positiva aqui apresentada. ´E tamb´em aqui proposta uma estrat´egia de recalibra¸c˜ao do controlador, a fim de melhorar o controlo do seguimento de referˆencia para o BIS dos pacientes, na presen¸ca de incertezas de modela¸c˜ao. Al´em disso, uma outra lei de controlo para a DoA ´e apresentada, composta por dois controladores em paralelo, cada um deles como proposto em Bastin and Provost (2002) para o controlo da massa total em sistemas compartimentais do tipo SISO. Ambos os controladores s˜ao baseados num modelo PPM. Esta lei de controlo alternativa permite a implementa¸c˜ao de uma estrat´egia para resolver o problema levantado pela existˆencia de incertezas nos parˆametros dos
desempenho do controlador.
A lei de controlo para sistemas positivos do tipo MISO foi avaliada em simula¸c˜ao e em ambiente cl´ınico. O seu desempenho foi considerado pelos anestesistas como satisfat´orio e potencial candidato para integra¸c˜ao num sistema de administra¸c˜ao per-sonalizada de f´armacos em anestesia geral.
Palavras-chave: Anestesia, Profundidade da Anestesia, BIS, Controlo Autom´atico, Sistemas Positivos
Abstract
In this work a positive control law is designed for multi-input positive systems that ensures asymptotic tracking of a desired output reference value. This control law can be viewed as a generalization of another one proposed in the literature for the control of the total mass in SISO compartmental systems, but is suitable for a wider class of positive systems. The controller proposed here is applied to the control of the depth of anesthesia (DoA), by means of the administration of propofol and remifentanil, when using a parameter parsimonious Wiener model (PPM), recently introduced in the literature. Its performance is illustrated by realistic simulations. A new estimation procedure for the parameters of the PPM is also presented. The fact that this model is parsimoniously parameterized allows a simple parameter estimation procedure, based on the patient’s step response. It is shown that the parameter estimates obtained in this way provide good results when used to tune the positive control law proposed here. A controller retuning strategy is also proposed in order to improve the BIS reference tracking in patients, in the presence of model uncertainties. For this purpose, the patient is simulated by the PK/PD Wiener model proposed in Marsh et al. (1991), Minto et al. (1997), and Schnider et al. (1998), while the controller is tuned for a PPM. Assuming that the misfit in the patient response is due to the existence of errors in the parameters of the PPM, a controller retuning strategy is obtained that improves the reference tracking, as is shown in the presented simulations. Moreover, another control law for the DoA is presented, which is composed by two controllers in parallel, each of them as proposed in Bastin and Provost (2002) for the control of the total mass in SISO compartmental systems. These controllers are based on a parameter
control law allows the implementation of a strategy to solve the problem raised by the existence of uncertainties in the parameters of the patients models. This technique significantly improves the performance of the controller.
The proposed control law for positive MISO systems was evaluated in simulations and in clinical environment. Due to its satisfactory performance the anesthesiologists con-sider it as a potential candidate for integration into a personalized drug administration system in general anesthesia.
Keywords: Anesthesia, Depth of Anesthesia, BIS, Automatic Control, Positive Sys-tems
Contents
Resumo v
Abstract vii
List of Tables xi
List of Figures xvi
1 Introduction 1
2 Output Reference Tracking 5
2.1 Problem Description . . . 6
2.2 Controller Design . . . 7
3 Depth Of Anesthesia 13 3.1 Model Description . . . 13
3.2 A Strategy Based on a Single Controller . . . 17
3.2.1 BIS Controller Design . . . 18
3.2.2 Assessment of the Controller Performance . . . 27 ix
3.2.4 A Strategy to Identify the Parameters of the PPM . . . 36
3.2.4.1 Identification Strategy Performance for DoA Control . 38 3.2.5 Retuning Strategy for DoA Control . . . 40
3.2.5.1 Retuning Strategy Performance . . . 43
3.2.6 State Observers . . . 48
3.2.6.1 Observer Performance . . . 55
3.3 A Strategy Based on Two Controllers in Parallel . . . 56
3.3.1 Paramenter Uncertainty . . . 62
3.3.2 Simulations . . . 64
3.4 DoA Controllers Comparison . . . 68
4 Clinical Cases 71 5 Conclusion 79 A Basic Definitions And Results 83 A.1 Stability Concepts . . . 84
A.2 Observers Overview . . . 85
B PK/PD Model Description 89
C Database 93
References 97
List of Tables
C.1 Patient features . . . 94 C.2 PPM parameters . . . 95 C.3 PK/PD model parameters - Hill eq. . . 96
List of Figures
3.1 Scheme for the BIS model. . . 14
3.2 Location of the eigenvalues of the matrix ¯A, for the different value
combinations of the parameters α, η and ρ within the ranges in this work, i.e., α ∈ [0.03 , 0.17], η ∈]0 , 5.70], ρ ∈ [0 , 500]. . . 24 3.3 Evolution of DoA, for ρ = 2, λ = 0.02 and λ = 1. . . 28 3.4 Evolution of DoA for different values of drugs proportion, i.e., ρ =
0, ρ = 2, ρ = 10 and for λ = 1. . . 29 3.5 Evolution of the dosage of propofol and of remifentanil, for λ = 1, ρ = 0
(without propofol infusion dose). . . 29 3.6 Evolution of the dosage of propofol and of remifentanil, for λ = 1, ρ = 2. 30 3.7 Evolution of the dosage of propofol and of remifentanil, for λ = 1, ρ = 10. 30 3.8 Evolution of DoA and of the dosages of propofol and of remifentanil,
assuming changes in the reference profiles (z∗ = 50 from the beginning
till t = 30 min, z∗ = 30 from t = 30 min till t = 60 min, and z∗ = 50
from then on), for λ = 1, ρ = 2. . . 31 3.9 Evolution of DoA in the presence of noise, for ρ = 2 and λ = 1. . . 31
istered by the anesthetist during the surgery (lower plot), along with the response of the corresponding PK/PD model to the same drug input. The average doses of propofol and of remifentanil reported were
4.21 mg min−1 and 0.0147 mg min−1 respectively. Notice that propofol
doses above 30 mg min−1 are not represented in the graphic. . . 33
3.11 BIS response of the Patient 15 and the infusion doses of propofol and of remifentanil obtained by the proposed control law, for z∗ = 50, ρ = 286, and λ = 10. The average doses of propofol and of remifentanil obtained
by the controller were respectively 4.27mg min−1and 0.0149mg min−1.
Notice that propofol doses above 30 mg min−1 are not represented in
the graphic. . . 34 3.12 BIS response of all eighteen simulated patients (by means of a PK/PD
model) controlled by the proposed control law, tuned for the
corre-sponding parameter parsimonious model, for z∗ = 50, ρ = 2, and λ = 10. 34
3.13 Evolution of the DoA, of Patient 9, and of the dosages of propofol and of remifentanil, assuming changes in the reference profiles (z∗ = 50 from
the beginning till t = 90 min, z∗ = 40 from t = 90 min till t = 120 min,
and z∗ = 50 from then on) and assuming changes in the value of the
drugs proportions (ρ = 0.2 in the first forty minutes and from then on it was changed to ρ = 3). λ = 1. . . 35 3.14 BIS evolution. In the first ten minutes the parameters are estimated
and in the following the BIS is controlled with the estimated parameters for tracking a BIS level of 50. . . 39 3.15 Comparison of BIS evolution of the patient controlled with the
pa-rameters estimated here (in red) and with the papa-rameters estimated in Mendon¸ca et al. (2012) (in blue). . . 40
3.16 BIS evolution of a simulated patient without applying the retuning strategy. The desired BIS level is set to be 50. . . 45 3.17 BIS evolution of a patient modeled by a PK/PD model with a retuning
strategy applied at time t = 20 min. The desired BIS level is set to be 50. . . 45 3.18 BIS evolution of a patient modeled by a PK/PD model with a retuning
strategy applied at time instants t1 = 20 min and t2 = 60 min. The
desired BIS level is set to be 50. . . 46 3.19 BIS evolution, in the presence of noise, of a patient modeled by a
PK/PD model with a retuning strategy applied instant t1 = 20 min.
The desired BIS level is set to be 50. . . 47 3.20 BIS evolution, in the presence of noise, of a patient modeled by a
PK/PD model with a retuning strategy applied at time instants t1 =
20 min and t2 = 70 min. The desired BIS level is set to be 50.
Parameters with uncertainty: γ and µ. . . 47 3.21 BIS evolution, in the presence of noise of a simulated patient, using an
observer. The reference value for the BIS level was set to be 50. . . 57 3.22 BIS evolution, in the presence of noise of a simulated patient, using an
observer and assuming changes in the reference profiles (z∗ = 50 from
the beginning till t = 50 min, z∗ = 40 from t = 50 min till t = 120 min,
and z∗ = 50 from then on). . . 57
3.23 Evolution of DoA, for λ = 1, ρ = 374, γ = ¯γ = 1, 88, µ = ¯µ = 1, 79,
and z∗ = 50. . . 65
3.24 Evolution of DoA, for λ = 1, ρ = 0, γ = ¯γ = 1, 88, and z∗ = 85. Here
the value of µ is irrelevant. . . 66 xv
in the first 3 minutes and, from then on, ρ = 374, γ = ˆγ = 1, 09,
µ = ¯µ = 1, 79, and z∗ = 50 versus the evolution of DoA, for λ = 1,
ρ= 374, γ = ¯γ = 1, 88, µ = ¯µ = 1, 79, and z∗ = 50. . . 67
3.26 Evolution of DoA, for λ = 1, ρ = 374, γ = ¯γ = 1, 88, µ = ¯µ = 1, 79, and
z∗ = 50 versus the evolution of DoA, for λ = 1, ρ = 0, γ = ¯γ = 1, 88,
and z∗ = 85 in the first 3 minutes, z∗ = 50, ρ = 374, γ = ˆγ = 1, 09,
µ = ¯µ = 1, 79 till retuning, and z∗ = 50, ρ = 374, γ = ˆγ = 1, 09,
µ= ˆµ = 2, 28 from then on. . . 67
3.27 In red is illustrated the BIS evolution of 18 simulated patients using two controllers in parallel. In blue is illustrated the BIS evolution of 18 simulated patients using one single controller. The desired BIS reference
value was set to be z∗ = 50. . . 69
4.1 Patient 1. The desired BIS level was set to be 50 during the all procedure. 74 4.2 Patient 2. The desired BIS level was initially set to be 50 and then was
set to be 45 until the end of the surgery. . . 75 4.3 Patient 3. The desired BIS level was set to be 50 during the all procedure. 76 4.4 Patient 4. The desired BIS level was initially set to be 50 and then was
set to be 45 until the end of the surgery. . . 77 4.5 Patient 5. The desired BIS level was initially set to be 50 and was
changed along the surgery, due to clinical conditions. . . 78
Chapter 1
Introduction
Positive systems are a particular class of systems that describe processes that involve intrinsically positive quantities, with applications in many areas of science and tech-nology. These systems naturally arise in modeling processes that occur in fields such as economy, biology, chemistry, ecology and population dynamics, communications, and medicine, among others.
These systems have gained increasing attention in the control literature during the last decades, due to their applicability and the theoretical challenges in controller design raised by the positivity constraint. In Luenberger (1979) a good introduction to modeling and analysis of positive systems is made, with applications to social science. Reference work on the structural properties and control of these systems may also be found in, for example, Farina (2002), Farina and Rinaldi (2000), Haddad et al. (2003), Roszak and Davison (2008), Roszak and Davison (2010),Kaczorek (2002), Willems (1976), Soltesz et al. (2012).
One of the applications of positive systems is the automatic control of drugs adminis-tration in patients during anesthesia, since the quantities involved in this process are all nonnegative.
General anesthesia enables a patient to tolerate surgical procedures that would other-1
wise inflict unbearable pain, potentiate extreme physiologic exacerbations, and result in unpleasant memories. The components of general anesthesia are areflexia (paraly-sis), hypnosis (unconsciousness and amnesia), and analgesia (absence of any sensation, including pain). The depth of anesthesia (DoA) is related to the intensity of these two latter components and is achieved by the administration of two drugs: a hypnotic and an analgesic. According to several studies (Tir´en et al. (2005), Grindstaff and Tobias (2004), Ekman et al. (2004), Wodey et al. (2005), Whyte and Booker (2003)) the DoA may be measured by means of the bispectral index (BIS). This index is a single dimensionless number, which is computed from the electroencephalogram (EEG) and ranges from 0 (equivalent to EEG silence) to 100 (equivalent to fully awake and alert). A BIS value between 40 and 60 is clinically desirable for general anesthesia purposes. This is usually achieved manually by the anesthesiologists. However, due to the high complexity of this procedure an automated system for drug administration would be a good support for the clinicians. This question has deserved the attention of several researchers and led to a number of contributions such as Dumont (2012), Ionescu et al. (2008), Mortier et al. (1998), Ionescu et al. (2011), Simanski et al. (2013), Bouillon (2008), but in the great majority of these contributions the control of the DoA is not fully automatic. It is precisely the challenge of developing a full automatic controller for the depth of anesthesia that motivates this thesis. This can be modeled as an output reference tracking problem. More specifically, after modeling the phenomenon in question by a control system in which the control input is the dosage of drugs to be administered and the output is the corresponding effect concentration, one seeks a control law that forces the system output to converge to the desired reference value. As a first step, we develop the necessary theoretical tools. To this end we consider single output positive systems with multiple inputs (MISO system) and design a nonlinear positive control law that ensures asymptotic tracking of a desired output reference value. This controller can be viewed as a generalization of the one proposed in Bastin and Provost (2002) for the control of the total mass in SISO compartmental positive systems. Moreover, our control law is suitable for a wider class of positive
3 systems.
The proposed controller is applied to the control of the depth of anesthesia by means of the hypnotic propofol and the analgesic remifentanil using a recently proposed parameter parsimonious Wiener model (PPM) for these drugs (see Silva et al. (2010)). The performance of the controller is analyzed by means of several simulations along with realistic simulations relying on identified real patients data collected in the surgery room during general anesthesia. The control law was also used for automatic administration of these two drugs to real patients during surgical procedures.
The concrete contributions of this thesis are the following:
• A nonlinear controller for automatic output reference tracking for MISO positive systems is designed allowing different convergence speed rates;
• Two different control strategies to control the depth of anesthesia in real patients during surgery, based on the PPM, are presented;
• Methods to identify parameters of the BIS model are suggested;
• A strategy to retune the proposed BIS controller in order to improve its accuracy is given;
• An observer to estimate the state of the BIS model based on the patient response during anesthesia procedure is obtained;
• Overall, a strategy to control the depth of anesthesia was achieved that con-tributes to the success of the personalized drug doses administration. Allowing different combinations of the dosages between the drugs administered as well as achieving different convergence rates.
This thesis is organized as follows. In Chapter 2, a control law is designed for output reference tracking in MISO positive systems. The application of the corresponding controller in general anesthesia is presented in the second section of Chapter 3, where
the performance of the proposed controller is illustrated by means of several simula-tions, and the results of its application in realistic simulated patients are presented. A procedure to identify the parameters of the parameter parsimonious model, a retuning strategy for DoA control to overcome the problems raised by modeling errors in modulation and a state observer design are also presented in the second section of this chapter. In the first section of Chapter 3, the BIS model is described and in the third section of the same chapter a new strategy to control the BIS of a patient is tested, where two controllers in parallel were used, in order to identify parameters of the nonlinear part of the PPM and also to control the DoA of a patient. In the last section of Chapter 3 a comparison between these two control strategies is made. The results obtained in the surgery room, by the use of the control law proposed in Section 3.2.1, are presented in Chapter 4. Conclusions follow in Chapter 5. Relevant concepts, definitions and results are collected in Appendix A.
Chapter 2
Output Reference Tracking For
MISO Positive Systems
In this chapter, a positive control law is designed for multi-input/single-output (MISO) positive systems that ensures asymptotic tracking of a desired output reference value. This control law can be viewed as a generalization of the one proposed in Bastin and Provost (2002) for the control of the total mass in SISO compartmental systems. However, whereas the control law in Bastin and Provost (2002) is only designed for compartmental systems, our control law is suitable for a wider class of positive systems as is sustained by the new theoretical results presented in this section.
For instance, in Soltesz et al. (2012) a controller by integral action together with a positivity constraint was proposed, showing that the input is positive provided that the integral gain is chosen sufficiently small. This controller does not presuppose a full knowledge of the model, however it does need the a priori knowledge of the steady-state gain matrix, in order to obtain a suitable value for the integral gain. This knowledge may take a long time to be obtained in case the process poles are not fast enough, which constitutes a drawback for its use in our application.
The controller presented here ensures reference tracking independently of the positive
value of the design parameter, without requiring the knowledge of the steady-state gain matrix. Nevertheless, it does require information about the model parameters and the corresponding state. However, these parameters can be identified in a short preliminary stage, and a state observer can be included in order to estimate the state online.
The control law, developed in this work, is considered to be nonlinear. This is not due to the use of nonlinear design methods, but rather because of the imposition of a positivity constraint on the control variable, which makes it a nonlinear function of the state of the system.
In Chapter 3, the controller proposed here is applied to the control of the depth of anesthesia (DoA).
2.1
Problem Description
Consider a positive system with m inputs and a single output. This system can be described by the state-space model
˙x(t) = Ax(t) + Bu(t) y(t) = Cx(t), (2.1)
where A is a n × n Metzler matrix, i.e., a matrix in which all the off-diagonal components are nonnegative, and B and C are matrices with nonnegative entries (see Godfrey (1983)) of dimension n × m and 1 × n, respectively. The restrictions on the system matrices guarantee that for nonnegative initial states x(0) and nonnegative inputs, the state and the output remain nonnegative. Here, for short, in the sequel (2.1) is denoted by (A, B, C). Moreover, for a vector v, the notations v ≥ 0 (v > 0) and v ≤ 0 (v < 0) mean that all its entries are respectively positive (strictly positive) and negative (strictly negative). The same applies to matrices.
2.2. CONTROLLER DESIGN 7 is sought such that the closed-loop system
˙x(t) = (A + BK)x(t) + BL y(t) = Cx(t), (2.2)
has bounded trajectories and tracks the reference, i.e., such that its output verifies
limt→∞y(t) = y∗.
2.2
Controller Design
Here we solve the problem of output reference tracking, by regarding it as a problem
of controlling the system to a level set Ωy∗ = {x ∈ Rn
+ : Cx = y
∗} in the state space,
where Rn
+= {x ∈ Rn: x ≥ 0}.
For this purpose, we first design an auxiliary control law, ˜u, and then impose positivity to ˜u in order to obtain a positive control input u. We also make the following assumptions: (A1) A is stable, (A2) CB is a nonzero row matrix and (A3) CA < 0.
Remark 2.2.1. Note that assumption A3 may be interpreted as a generalization of one of the features of compartmental systems, where the matrix A is diagonally dominant by columns. In fact, a Metzler square matrix A = (aij), i, j = 1, . . . , n, is diagonal
dominant by columns if |ajj| > n X i=1;i6=j aij, j = 1, . . . , n. (2.3)
In a compartmental system the diagonal elements ajj are non positive and, hence, the
previous inequality may be written as Pn
i=1aij < 0, j = 1, . . . , n, or, equivalently,
as
1 · · · 1 A < 0, which is a particular case of the condition CA < 0 for
C =
1 · · · 1.
Let
˜u(t) = −ECAx(t) + Eλ(y∗− y(t)), (2.4)
where λ > 0 is a design parameter, and E is a column matrix with nonnegative entries such that CBE = 1. Note that such a matrix always exists since CB has nonnegative entries, at least one of each is strictly positive. The application of this control input leads to the closed-loop dynamics
˙x(t) = Ax(t) + B(Eλ(y∗ − y(t)) − ECAx(t)) (2.5)
which implies that
˙y(t) = C ˙x(t) = CAx(t) + λ(y∗− y(t)) − CAx(t) (2.6)
= −λ(y(t) − y∗) (2.7)
and therefore
˙ d
y(t) − y∗ = −λ(y(t) − y∗). (2.8)
Hence, y(t) − y∗ = e−λt(y(0) − y∗) and
lim
t→∞y(t) = y
∗
, (2.9)
which means that the output reference value is asymptotically tracked.
In the sequel it is shown that reference tracking can still be achieved even when a positivity restriction to the control input is imposed. This positivity restriction is made
2.2. CONTROLLER DESIGN 9
componentwise and corresponds to taking the control input as u = u1 · · · um
T
with ui = max(˜ui,0), where ˜ui denotes the i − th component of ˜u. Note that ˜ui =
Ei(−CAx + λ(y∗ − y)), where Ei ≥ 0 is the i − th entry of E. Therefore if ˜ui < 0
then −CAx + λ(y∗− y) < 0, and all the other components ˜u
j of ˜u corresponding to
nonzero Ej are negative as well. This allows to conclude that either u = ˜u or u = 0.
In this latter case
λ(y∗− y) < CAx. (2.10)
Since, by assumption (A3), CA < 0 and x ≥ 0, then CAx ≤ 0 and (2.10) implies
that y∗− y <0.
To prove that all trajectories converge to y∗, we apply the LaSalle’s invariance principle
(see LaSalle (1976), Leine and Wouw (2008), Liao et al. (2007), Bullo (2005)) to the Lyapunov function
V(x) = 1
2(y
∗− y)2 (2.11)
for the system (2.1) on Rn
+. For u = ˜u:
˙V (x) = −(y∗− y) ˙y = −(y∗− y)C ˙x (2.12)
= −(y∗− y)C (Ax + B [Eλ(y∗− y) − ECAx]) (2.13)
= −λ(y∗− y)(y∗− y) = −λ(y∗− y)2 ≤ 0 (2.14)
For u = 0 (which can only happen when y∗ − y <0, see (2.4)):
˙V (x) = −(y∗ − y)C ˙x = − (y∗− y) | {z } <0 CAx | {z } ≤0 ≤ 0 (2.15)
Thus ˙V (x) = −λ(y∗− y)2 for u = ˜u −(y∗− y)CAx for u = 0 (2.16)
By the LaSalle’s invariance principle, all system trajectories converge to the largest set contained in
W = {x ∈ Rn+ : ˙V (x) = 0}, (2.17)
which is forward-invariant under the closed-loop dynamics. It follows from (2.16) that
˙V (x) = 0 either when u = ˜u and y = y∗ or when u = 0,which implies y∗ < y, and
CAx= 0. So we get
W = {x ∈ Rn+ : y = y∗ or (y∗ < y and CAx = 0)}. (2.18)
Moreover, the set Ωy∗ of positive states for which the corresponding output y equals
y∗ is forward-invariant under the closed-loop dynamics. In fact, let F (x) be the vector
field associated with the closed-loop system
˙x = Ax + Bu u= max(˜u, 0). (2.19)
When Cx = y∗, u = ˜u = −ECAx ≥ 0 and F (x) = Ax − BECAx. As a consequence,
recalling that CBE = 1, one obtains CF (x) = 0 showing that F (x) is tangent to Ωy∗.
So Ωy∗ is forward-invariant under the closed-loop dynamics.
On the other hand, the trajectories starting in a state x for which Cx > y∗ and
2.2. CONTROLLER DESIGN 11
Cx > y∗ ⇒ u= 0 (2.20)
⇒ ˙x(t) = Ax(t). (2.21)
Since A is assumed to be stable, if the control would remain equal to zero, then
limt→∞Cx(t) would be zero. This implies that at a certain time instant, say t∗, Cx(t∗)
reaches the value Cx(t∗) = y∗, i.e. x(t∗) ∈ Ω
y∗. From this instant on the trajectories
remain indefinitely in the forward-invariant set Ωy∗. Therefore, the largest invariant
subset contained in W is Ωy∗ and, by LaSalles’s invariance principle, all the
closed-loop system trajectories converge to this set, which means that limt→∞y(t) = y∗ as
desired.
The study just developed leads to the following result.
Theorem 2.2.1. Let(A, B, C) be a positive MISO linear system, such that A is stable, CA < 0 and CB 6= 0. If u = max(˜u, 0), ˜u = −ECAx + Eλ(y∗ − y), with λ > 0, and E ≥ 0 such that CBE = 1, then the closed-loop system output y(t) verifies
Chapter 3
Depth Of Anesthesia: Control And
Identification
Combinations of drugs are used in general anesthesia because no single drug is able to provide all its necessary components (namely, analgesia, hypnosis, areflexia) without seriously compromising hemodynamic and/or respiratory function, impairing oper-ating conditions, or delaying postoperative recovery. Ideal combination of dosing facilitates optimal therapeutic effect without producing significant side effects.
Here, two different control schemes are designed for the automatic administration of the hypnotic agent propofol and the opioid analgesic remifentanil to patients during surgery, in order to achieve a desired level of unconsciousness. This level is measured in terms of the depth of anesthesia (DoA), here denoted by z(t), which is a feature that can be related to the quantities of administered drugs as explained next.
3.1
Model Description
In what concerns the DoA, the response to the administration of the hypnotic drug propofol and of the analgesic drug remifentanil is commonly modeled as a high
der pharmacokinetic/pharmacodynamic (PK/PD) Wiener model (Bailey and Haddad (2005), Schnider et al. (1998), Marsh et al. (1991), Minto et al. (1997), Minto et al. (2000)). This model has a static nonlinear part, given by the Hill equation (Minto et al. (2000)), preceded by a compartmental linear part that depends on several parameters, corresponding to compartment volumes, micro-constant rates for mass transfer between compartments and drug clearance in the human body (see Figure 3.1). This is an undesirable situation, as some of the parameters are unidentifiable. For this reason, mean values of the parameters are often considered. However, this does not always lead to adequate models, for instance, for control purposes. To overcome this situation, a new parameter parsimonious PK/PD Wiener model (PPM), with a reduced number of parameters, describing the joint effect of propofol and of remifentanil has been introduced in Silva et al. (2010), allowing a less complex parameter identification procedure. The main goal of this chapter is to design a controller for the DoA based on this parameter parsimonious Wiener model.
Figure 3.1: Scheme for the BIS model.
The effect concentration of propofol (cp
e) and of remifentanil (cre) can be modeled by
the parameter parsimonious Wiener model developed by Silva et al. (2010). According to this model
3.1. MODEL DESCRIPTION 15 cpe(s) = Hp(s) = k1k2k3α 3 (k1α+ s)(k2α+ s)(k3α+ s) up(s), (3.1) cre(s) = Hr(s) = l1l2l3η 3 (l1η+ s)(l2η+ s)(l3η+ s) ur(s), (3.2) where cp
e(s), cre(s) denote the Laplace transforms of cpe(t) and cre(t), respectively, and
up(s) and ur(s) are the Laplace transforms of the administered doses of propofol, up(t),
and of remifentanil, ur(t), in mg min−1. Each of the transfer functions in (3.1) and
(3.2) has three aligned poles, more concretely, the first one has poles (−k1, −k2, −k3)α
and the second one has poles (−l1, −l2, −l3)η. The parameters kj, lj, j = 1, 2, 3 were
chosen in Silva et al. (2010) according to the collected patient data and fixed at the
values k1 = 10, k2 = 9, k3 = 1, l1 = 3, l2 = 2, l3 = 1. The parameters α and η are
patient dependent.
The joint effect of the concentrations of propofol and remifentanil on the BIS level is modelled in Silva et al. (2010) by the generalized Hill equation:
z(t) = z0
1 + (µUp+ Ur)γ, (3.3)
where µ, γ are patient dependent parameters, z0is the effect at zero concentration, and
Up and Ur respectively denote the potencies of propofol and remifentanil, which are
obtained by normalizing the effect concentrations with respect to the concentrations
that produce half the maximal effect when the drug acts isolated (denoted by ECp
50 and ECr 50, respectively), i.e.: Up = c p e EC50p and U r = c r e ECr 50 . (3.4)
In this work we consider α ∈ [0.03 , 0.17], η ∈]0 , 5.70], ECp
ECr
50 = 0.01 mg/ml. The values of EC
p
50 = 10 mg/ml and EC50r = 0.01 mg/ml are
taken from Mendon¸ca et al. (2012) and the intervals for the values of α and η are
obtained as follows: α ∈ [¯α − 2σα , ¯α + 4σα] and η ∈]0 , ¯η + 4ση], where ¯α and σα
respectively denote the mean and standard deviation of the parameters identified in
Mendon¸ca et al. (2012), and ¯η and σηhave the obvious meaning, now for the parameter
η. The choice of lower bound 0 for η is due to the fact that η should be positive and
¯η − 2ση <0.
The transfer functions in (3.1) and (3.2) can be represented by the following state-space model: ˙xi = Aixi+ Biui ci e = 0 0 1 xi, i= p, r (3.5) where xi = xi 1 xi2 xi 3 is the state, Ap = −10α 0 0 9α −9α 0 0 α −α , Ar = −3η 0 0 2η −2η 0 0 η −η , Bp = 10α 0 0 and Br = 3η 0 0 . (3.6)
3.2. A STRATEGY BASED ON A SINGLE CONTROLLER 17
Defining U = µUp+ Ur yields
z(t) = z0 1 + Uγ, (3.7) with U = 1 EC50p µc p e+ 1 ECr 50 cre= 0.1µcpe+ 100cre. (3.8)
This leads to the following model
˙x(t) = Ax(t) + Bu(t) U(t) = 0.1µcp e(t) + 100cre(t) = Cx(t) z(t) = z0 1+Uγ, (3.9) where x(t) = xp(t) xr(t) , A = Ap 0 3×3 03×3 Ar , B = Bp 0 3×1 03×1 Br , C = 0 0 0.1µ 0 0 100 . (3.10)
3.2
A Strategy Based on a Single Controller
In this section, a control law based on the Theorem 2.2.1 is developed in order to control the depth of anesthesia in patients, by means of the administration of propofol and of remifentanil. Our results prove to be useful for the control of the depth of anesthesia, a problem that has lately deserved much attention. For instance, the work developed in Soltesz et al. (2012) presents two controllers in parallel for the DoA of a patient based on a PID controller for the administration of propofol and a proportional controller for the input of remifentanil. Here, we present one single multi-output feedback controller for both drugs.
3.2.1
BIS Controller Design
As mentioned before, for surgery purposes it is desirable to maintain the BIS close to
a certain reference level zref between 40 and 60. This can be achieved by designing a
control law that forces U(t) to follow the constant reference level
Uref = γ
r z
0
zref −1. (3.11)
In order to apply the design method of the previous subsection, the product CB should
be a nonzero row. However, here CB = 0 0
and the condition is not met. To overcome this problem, instead of the output U(t), the output
M(x(t)) = 0.1Mp(xp(t)) + 100Mr(xr(t)) = CMx(t), (3.12) with Mp(xp) = 1 1 1 xp, Mr(xr) = 1 1 1 xr and CM = 0.1 0.1 0.1 100 100 100 , (3.13)
is considered. The notations M(x(t)), Mp(xp(t)), and Mr(xr(t)) are inspired by the
fact that, in compartmental systems, the sum of the state components usually cor-responds to the total substance mass present in the system. As we shall later see,
a connection between the reference value Uref and an adequate reference value Mref
for M can be established, in such a way that when limt→∞M(x(t)) = Mref then
limt→∞U(t) = Uref and limt→∞z(t) = zref, as desired.
In a first stage, we show that every positive constant reference value M∗ for M (x(t))
3.2. A STRATEGY BASED ON A SINGLE CONTROLLER 19
Then, in a second step, we determine which value should be taken for M∗ in order to
ensure that the desired constant reference value zref for the BIS level is achieved.
Since now, for the new output matrix CM, defined in (3.13), CMB =
α 300η
is nonzero, the applicability of the proposed controller design method is guaranteed.
Proposition 3.2.1. Let (A, B, CM) be a positive MISO linear system, with A, B as
in (3.10) and CM as in (3.13). Define E = ρ 1 1 αρ+ 300η, (3.14)
where ρ >0 is an arbitrary nonnegative value. Then, applying the control law
u= max(0, ˜u), (3.15) with ˜u = ˜up ˜ur = E (−CMAx+ λ(M ∗− M)) and λ > 0 (3.16) = ρ 1 (−CMAx+ λ(M ∗− M)) 1 αρ+ 300η | {z } ¯ u , (3.17) and λ >0
to the system (A, B, CM), yields limt→∞M(x(t)) = M∗.
Proof. Since CMBE = 1, all the conditions of Theorem 2.2.1 are satisfied yielding the
Remark 3.2.1. Note that, in this control law, λ and ρ are design parameters. More-over ˜up ˜ur = ρ 1
¯u, with ¯u as in (3.17), meaning that the proportion of propofol
and remifentanil is ρ : 1.
Since the choice of the positive parameter ρ does not affect the tracked reference value, this may constitute an advantage, as it allows one to choose the proportion between the two drugs, in order to accommodate clinical restrictions or considerations, without significant consequences in terms of the effect. This will be illustrated later on in the simulations.
In order to determine which value of Mref should be chosen for M∗ in the control
law (3.17) to guarantee that the BIS level z(t) tracks a desired constant value z∗,
an analysis is made of the values U(t) obtained for the closed-loop system. For this
purpose it will be first proved that the state of the closed-loop system (A, B, CM)
with the control law (3.15) converges to an equilibrium point x∗. To show this, since,
as mentioned in the previous section, the trajectories x(t) converge to the
forward-invariant set ΩM∗ = {x ∈ R6
+ : M(x) = M
∗}, it is enough to prove that, for the
restriction of the closed-loop system to this set, the state trajectories converge to an
equilibrium point x∗. Note that this restriction is well defined as Ω
M∗ is
forward-invariant under the closed-loop dynamics.
In order to obtain the dynamics of the system restricted to ΩM∗, note that when
x ∈ ΩM∗, i.e., when M(x) = M∗, the control ˜u is given by ˜u = −ECMAx, and the
closed-loop system can be described as
˙x = ˜Ax (3.18)
3.2. A STRATEGY BASED ON A SINGLE CONTROLLER 21 ˜ A = A − BECMA = a1−10α 8a1 a1 a2 a2 a2 9α −9α 0 0 0 0 0 α −α 0 0 0 a3 8a3 a3 a4−3η a4 a4 0 0 0 2η −2η 0 0 0 0 0 η −η where a1 = α2ρe, a2 = 1000αηρe, a3 = 0.3αηe, a4 = 300η2e, e= αρ+300η1
and A, B, CM and E are given by (3.10), (3.13), and (3.14).
In order to analyze the stability of the closed-loop system restricted to ΩM∗, the
evolution equations of this restriction are next obtained.
When M(x(t)) = M∗,
xp1(t) = 10M∗− xp2(t) − xp3(t) − 1000(xr1(t) + xr2(t) + xr3(t)). (3.19)
Replacing this in the equation ˙x(t) = ˜Ax(t) yields
˙xp 1(t) = − ˙x p 2(t) − ˙x p 3(t) − 1000( ˙xr1(t) + ˙xr2(t) + ˙xr3(t)) ˙xp 2(t) = 9α(10M∗− x p 2(t) − x p 3(t) − 1000(xr1(t) + xr2(t) + xr3(t))) − 9αx p 2(t) ˙xp 3(t) = αx p 2(t) − αx p 3(t) ˙xr 1(t) = 10a3M∗+ 7a3x p 2(t) + (a4−1000a3−3η)xr1(t)+ +(a4−1000a3)(xr2(t) + xr3(t)) ˙xr 2(t) = 2ηxr1(t) − 2ηxr2(t) ˙xr 3(t) = ηxr2(t) − ηxr3(t). (3.20)
Since the state component xp
1(t) can be obtained from the others state components
(see (3.19)), we can conclude that the closed-loop system dynamics restricted to ΩM∗
is described by the evolution of the vector
¯x = xp2 xp3 xr1 xr 2 xr3 , (3.21)
which is given by the last five equations from (3.20). These equations can be written in matrix form as:
˙¯x = ¯A¯x + R, (3.22) with ¯ A = −18α −9α −9000α −9000α −9000α α −α 0 0 0
7a3 0 a4 −1000a3−3η a4−1000a3 a4−1000a3
0 0 2η −2η 0 0 0 0 η −η and R = 90αM∗ 0 10a3M∗ 0 0 .
Moreover, equation (3.22) can be written in the form: ˙
d
¯x − ¯x∗= ¯
3.2. A STRATEGY BASED ON A SINGLE CONTROLLER 23 where ¯x∗ = ρ ρ 1 1 1 M∗ 3(0.1ρ + 100) (3.24)
is the only equilibrium point of (3.22), i.e., the only point such that ¯x(t) ≡ ¯x∗ satisfies
˙¯x = ¯A¯x(t) + R, as ˙¯x(t) = 0 and ¯A¯x∗ = −R. This can be verified by performing the
corresponding multiplication ¯A¯x∗ and checking that the result equals −R. Moreover,
the uniqueness of ¯x∗ follows from the invertibility of ¯A for the considered parameter
range.
As is shown in Fig. 3.2, where the locations of the eigenvalues of ¯A are plotted for
the different value combinations of the parameters α and η, within the ranges in this
paper, and for ρ ∈ [0 , 500], the matrix ¯Ais stable, i.e., all its eigenvalues have negative
real part. The restriction ρ ∈ [0 , 500] is only imposed to limit the high computation level, however this feat is not a limitation to the applicability of the controller here proposed, since the ρ values observed in the collected clinical cases are included in
the considered interval. Thus, ¯x(t) converges to ¯x∗ = (ρ, ρ, 1, 1, 1) M∗
3(0.1ρ+100). Recalling
that equation (3.19) holds, this implies that x(t) = (xp∗
1 , ¯x ∗) with xp∗1 = 10M∗− 1 1 1000 1000 1000 ¯x∗ (3.25) = 3(0.1ρ + 100)ρM∗ . (3.26) Thus x(t) converges to x∗ = (ρ, ρ, ρ, 1, 1, 1) M ∗
3(0.1ρ + 100) under the closed-loop
Figure 3.2: Location of the eigenvalues of the matrix ¯A, for the different value combinations of the parameters α, η and ρ within the ranges in this work, i.e.,
α ∈[0.03 , 0.17], η ∈]0 , 5.70], ρ ∈ [0 , 500].
Recalling that U(t) = 0.1µxp
3(t) + 100xr3(t), one concludes that, under the closed loop
dynamics, U∗ = lim t→∞U(t) = 0.1µ limt→∞x p 3(t) + 100 limt→∞x r 3(t) (3.27) = 0.1µ3(0.1ρ + 100)ρM∗ + 1003(0.1ρ + 100)M∗ (3.28) = M∗ 0.1µρ + 100 3(0.1ρ + 100). (3.29)
Consequently, in order to track a constant reference level Uref for U(t) it is enough to
take
M∗ = Mref = 3(0.1ρ + 100)
0.1µρ + 100 Uref (3.30)
3.2. A STRATEGY BASED ON A SINGLE CONTROLLER 25 Together with expression (3.11), these considerations lead to the following result (see Nogueira et al. (2014a)).
Proposition 3.2.2. Let (A, B, CM) be a positive MISO linear system, with A, B
as in (3.10) and CM as in (3.13). Then, applying the control law u as in (3.15),
with M∗ = Mref = 3(0.1ρ+100)0.1µρ+100 qγ z0
zref −1, to the system (A, B, CM), achieves tracking
of the constant reference value zref for the BIS level z(t) defined as in (3.3), i.e.,
limt→∞z(t) = zref.
Proof. It follows from what has just been said that taking in (3.15)
M∗ = Mref = 3(0.1ρ + 100) 0.1µρ + 100 γ r z 0 zref −1
achieves tracking of the constant reference value for U: Uref = γ
r z
0
zref −1.
The result now follows from expression (3.11).
As mentioned before, the controller presented in the Proposition 3.2.2 allows to fix
a proportion ρ between the doses of propofol and of remifentanil, i.e., ρ = up
ur
. However, if we desire to administer only propofol into the patient we have to make some adjustments in the designed control law. For this purpose, we follow the same reasoning we used to achieve the results presented in Proposition 3.2.2, but instead
of fixing the proportion ρ = up
ur
, we fix the proportion ρ2 =
ur
up
, that is, we fix the proportion between the doses of remifentanil and of propofol. With this, we obtain the following result.
Proposition 3.2.3. Let (A, B, CM) be a positive MISO linear system, with A, B as
˜u = ˜up ˜ur (3.31) = 1 ρ2 (−CMAx+ λ(M ∗− M)) 1 α+ 300ηρ2 , (3.32) with λ >0, ρ2 > 0, M∗ = Mref = 3(0.1 + 100ρ2) 0.1µ + 100ρ2 γ r z 0
zref −1, to the system (A, B, CM),
achieves tracking of the constant reference value zref for the BIS level z(t) defined as
in (3.3), i.e., limt→∞z(t) = zref.
Now, if we intend to administer only propofol into the patient we may use the control
law as designed in Proposition 3.2.3 with ρ2 = 0.
Both Proposition 3.2.3 and 3.2.2 may be generalized by the next proposition.
Proposition 3.2.4. Let (A, B, CM) be a positive MISO linear system, with A, B as
in (3.10) and CM as in (3.13). Then, applying the control law u= max(0, ˜u) where,
˜u = ˜up ˜ur (3.33) = ρ1 ρ2 (−CMAx+ λ(M ∗− M)) 1 αρ1+ 300ηρ2 , (3.34) with M∗ = Mref = 3(0.1ρ1 + 100ρ2) 0.1ρ1µ+ 100ρ2 γ r z 0 zref −1, λ > 0, (ρ1 ≥ 0 and ρ2 = 1) or
(ρ1 = 1 and ρ2 ≥ 0) to the system (A, B, CM), achieves tracking of the constant
reference value zref for the BIS level z(t) defined as in (3.3), i.e., lim
3.2. A STRATEGY BASED ON A SINGLE CONTROLLER 27
Without loss of generality, from now on, we will consider the case where ρ2 = 1 and
ρ= ρ1 =
up
ur.
As stated in Proposition 3.2.4, once the values of γ and µ are known for a particular
patient, the value of Mref to be considered in the control law is:
Mref = 3(0.1ρ1+ 100ρ2) 0.1ρ1µ+ 100ρ2 γ r z 0 zref −1, (3.35)
where z0 = 97.7 and zref is the desired BIS level. Here it is assumed that the values
of γ and µ are available, however, if this is not the case, these values can be identified in a first stage, before turning on the controller. Another possibility is to start with
an average value for γ and µ and hence M∗ as a first estimate and then retune this
value after γ and µ are more accurately identified. A similar procedure was used in Nogueira et al. (2013a). Here, it is also assumed that the values of the parameters
α, η are known. This does not happen in practice. However, in a real situation the
controller action only starts after an initial drug bolus is given. This is a current practice with other automatic controllers previously used in clinical environment and allows to identify the parameters of the patient model.
3.2.2
Assessment of the Controller Performance
In this subsection the performance of the control law presented in the previous sections
is assessed by means of several simulations. For this purpose, we consider: z0 = 97.7;
α = 0.0759; η = 0.583; µ = 1.79; γ = 1.88. These parameter values are the mean
values used in the work developed in Mendon¸ca et al. (2012), to which we refer for further explanation. We also assume that it is intended that the DoA of a patient corresponds to the reference value of 50 for the BIS signal. By (3.11), this means that
U must follow the reference U∗ = 0.9753. Once these values are fixed, the control law
the speed of convergence to the reference value, as can be seen in Fig. 3.3 where the values λ = 0.02, λ = 1 were respectively taken for a fixed value of ρ = 2 (corresponding to proportion of 2 : 1 for propofol and remifentanil).
0 2 4 6 8 10 50 55 60 65 70 75 80 85 90 95 100 Time[min] D o A m ea su re m en t (B IS ) λ = 1 λ = 0.02
Figure 3.3: Evolution of DoA, for ρ = 2, λ = 0.02 and λ = 1.
On the other hand the parameter ρ specifies the desired proportion between the administered amounts of propofol and remifentanil, which may be chosen according to clinical criteria. Figure 3.4 illustrates DoA effect for different drug proportions (namely, ρ = 0, ρ = 2, and ρ = 10) and Figures 3.5, 3.6, and 3.7 present the evolution of the corresponding drug dosages. As can be seen in Fig. 3.4, in spite of the variation of the drug proportion (ρ) the desired effect is practically the same. It turns out that this may constitute an advantage, since the choice of the proportion can be made in order to accommodate individual clinical restrictions or considerations, without significantly consequences in terms of the effect.
Figure 3.8 illustrates the performance of the control algorithm in the presence of a change of the reference profile. In the first thirteen minutes it is intended that the BIS follows the reference 50, in the following thirteen minutes the BIS reference level is set to 30 and in the last twenty minutes the BIS should again follow the reference level 50. It may be seen that the controller has a good performance also in this case. As expected, when the reference decreases there is a small bolus of propofol and remifentanil, in order to follow the reference, and when it increases no drug is
3.2. A STRATEGY BASED ON A SINGLE CONTROLLER 29 0 2 4 6 8 10 50 55 60 65 70 75 80 85 90 95 100 Time[min] D o A m ea su re m en t (B IS ) ρ = 10 ρ = 2 ρ = 0
Figure 3.4: Evolution of DoA for different values of drugs proportion, i.e., ρ = 0, ρ = 2, ρ = 10 and for λ = 1. 0 5 10 15 20 −0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 Time[min] D ru g In fu si o n R a te [m g m in − 1] Propofol infusion Remifentanil infusion
Figure 3.5: Evolution of the dosage of propofol and of remifentanil, for λ = 1, ρ = 0 (without propofol infusion dose).
administered.
In a real clinical environment, the measurements of the patients BIS levels usually present a high level of noise, due to the nature of the sensors and the physiological inherent signs. This fact implies the existence of noise in the estimation of the states in the corresponding theoretical model. In Fig. 3.9 the evolution of the DoA of a patient
0 5 10 15 20 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Time[min] D ru g In fu si o n R a te [m g m in − 1] Propofol infusion Remifentanil infusion
Figure 3.6: Evolution of the dosage of propofol and of remifentanil, for λ = 1, ρ = 2.
0 5 10 15 20 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Time[min] D ru g In fu si o n R a te [m g m in − 1] Propofol infusion Remifentanil infusion
Figure 3.7: Evolution of the dosage of propofol and of remifentanil, for λ = 1, ρ = 10.
is illustrated in the presence of noise in the measurement of the BIS level, in order to mimic the real data collected in the surgery room. For this purpose, Gaussian white
noise with zero mean and standard deviations σnoise= 3 was added to the output. As
we can see, in spite of the presence of noise in the BIS measurements, the behavior of the controlled output of the patient is clinically accepted.
3.2. A STRATEGY BASED ON A SINGLE CONTROLLER 31 0 10 20 30 40 50 60 70 80 20 40 60 80 100 Time[min] D oA m ea su re m en t (B IS ) BIS evolution Reference value for BIS
0 10 20 30 40 50 60 70 80 0 0.1 0.2 0.3 0.4 Time[min] D ru g In fu si on R at e [m g m in − 1] Propofol infusion Remifentanil infusion
Figure 3.8: Evolution of DoA and of the dosages of propofol and of remifentanil,
assuming changes in the reference profiles (z∗ = 50 from the beginning till t = 30 min,
z∗ = 30 from t = 30 min till t = 60 min, and z∗ = 50 from then on), for λ = 1, ρ = 2.
0 50 100 150 30 40 50 60 70 80 90 100 Time[min] Do A m ea su re m en t (B IS ) BIS evolution
Figure 3.9: Evolution of DoA in the presence of noise, for ρ = 2 and λ = 1.
3.2.3
A Realistic Simulation Study
As a necessary preliminary step before implementation in clinical environment, the performance of the controller proposed in this work is tested on a bank of realistic simulated patients. These patients are simulated by means of PK/PD Wiener models (see Marsh et al. (1991), and Minto et al. (1997)) obtained from real data collected
during eighteen breast surgeries. The patients (all female, with 54 ± 13 years of age, a height of 160 ± 5cm, and 69 ± 18Kg) were subject to general anesthesia under propofol and remifentanil administration. The DoA was monitored by the BIS and was manually controlled around clinically accepted values by the anesthetist. Alaris GH pumps were used to administer both drugs, propofol and remifentanil. Infusion rates, BIS values and other physiological variables were acquired every five seconds (Mendon¸ca et al. (2012)).
For each patient, the corresponding PK/PD Wiener model was obtained as follows. The parameters of the linear part were computed according to Marsh et al. (1991), Minto et al. (1997), and Schnider et al. (1998) based on the relevant patient charac-teristics (see Appendix B), whereas the parameters of the nonlinear part (generalized Hill equation) were identified from the surgery data, by Mendon¸ca et al. (2012). In order to test the proposed control strategy, each simulated patient is also modeled by the parsimonious parameter Wiener model of Silva et al. (2010), with parameters identified as in Mendon¸ca et al. (2012), and a control law of the form (3.17) tuned for that model is applied to the simulated patient. The set of parameter values identified by Mendon¸ca et al. (2012) is given in Appendix C, where also the corresponding relevant patient features that allow to compute the PK/PD model parameters are displayed.
Here we present the results obtained for Patients 9 and 15 of the database in two different simulations settings respectively. Patient 9 is a woman, with 51 years of age, a height of 163 cm, and with 55 kg. The corresponding identified parameters
were: ECp
50 = 12.17 mg/ml, EC50r = 0.031 mg/ml, γ = 1.86, µ = 3.84, α = 0.07,
and η = 0.28. Patient 15 is a woman, with 73 years of age, a height of 160 cm, a
weight of 75 kg, for which the identified parameters were: ECp
50 = 12.41 mg/ml,
ECr
50= 0.016 mg/ml, γ = 1.68, µ = 3.47, α = 0.10, and η = 0.04.
Figure 3.10 shows the BIS response of the patient 15 to the drug doses administered by the anesthetist during the surgery, along with the response of the corresponding PK/PD model to the same drug input. The similarity between the two responses
3.2. A STRATEGY BASED ON A SINGLE CONTROLLER 33 supports the idea of using the identified PK/PD models in order to simulate real patients. 0 20 40 60 80 100 120 140 160 180 20 40 60 80 100 D o A m ea su re m en t (B IS ) Time [min]
BIS evolution of patient 15 during surgery BIS evolution of patient 15 modeled by PK/PD model
0 20 40 60 80 100 120 140 160 180 0 10 20 30 Time[min] D ru g In fu si o n R a te [m g m in − 1]
Propofol infusion during surgery Remifentanil infusion during surgery
Figure 3.10: BIS response of the Patient 15 (upper plot) to the drug doses administered by the anesthetist during the surgery (lower plot), along with the response of the corresponding PK/PD model to the same drug input. The average doses of propofol
and of remifentanil reported were 4.21 mg min−1 and 0.0147 mg min−1 respectively.
Notice that propofol doses above 30 mg min−1 are not represented in the graphic.
In Fig. 3.11 the BIS response of the simulated patient 15 is presented, where the DoA control was performed using the control law (3.17) for tracking a BIS level of 50; this is an approximation value of the average real BIS obtained throughout the surgery. The selected proportion between the two drugs was the corresponding average reported in the monitored real case, ρ = 286. The average of the doses of propofol
and of remifentanil obtained by the controller were respectively 4.27 mg min−1 and
0.0149 mg min−1; these values are quite close to the reported real ones, 4.21 mg min−1
and 0.0147 mg min−1 respectively. The obtained set-point for the BIS level was 54.7,
presenting an relative error of only 9%.
Figure 3.12 illustrates the application of the proposed control law to all the 18 sim-ulated patients, for a desired reference BIS level of 50 (the remaining parameters are described in the caption of the figure). As it can be seen, the BIS level obtained in all
0 20 40 60 80 100 120 140 160 40 60 80 100 Time[min] D o A m ea su re m en t (B IS )
BIS evolution of the simulated patient 15
0 20 40 60 80 100 120 140 160 0 10 20 30 Time[min] D ru g In fu si o n R a te [m g m in − 1] Propofol infusion Remifentanil infusion
Figure 3.11: BIS response of the Patient 15 and the infusion doses of propofol and of remifentanil obtained by the proposed control law, for z∗ = 50, ρ = 286, and λ = 10. The average doses of propofol and of remifentanil obtained by the controller were
respectively 4.27 mg min−1 and 0.0149 mg min−1. Notice that propofol doses above
30 mg min−1 are not represented in the graphic.
the cases is very close to the desired one.
0 20 40 60 80 100 120 140 160 0 10 20 30 40 50 60 70 80 90 100 D o A m ea su re m en t (B IS ) Time [min]
BIS evolution of the 18 simulated patients
Figure 3.12: BIS response of all eighteen simulated patients (by means of a PK/PD model) controlled by the proposed control law, tuned for the corresponding parameter
3.2. A STRATEGY BASED ON A SINGLE CONTROLLER 35 In Fig. 3.13 the performance of the control in the presence of a change of the reference profile and in the value of drugs proportion is illustrated (for patient 9). During the first ninety minutes it is intended that the BIS follows a reference value of 50, in the following thirty minutes the BIS reference level is set to be 40 and in the last forty minutes the BIS is regulated to follow again the reference level of 50. As expected, the decrease in the reference value produces a peak in the doses of propofol and remifentanil, whereas when the reference value increases no drug is administered. The proportion between the two drugs was ρ = 0.2 in the first forty minutes and from then on it was changed to ρ = 3. In spite of the variation of the drug proportion the BIS follows the desired reference. The possibility of changing the drug proportion during the surgery without affecting the reference tracking results constitutes a considerable advantage. Indeed, clinical practice has often shown the need of adjusting the dosage profile according to the overall physiological response of the patient.
0 20 40 60 80 100 120 140 160 40 60 80 100 Time[min] D o A m ea su re m en t (B IS )
BIS evolution of the simulated patient 15
0 20 40 60 80 100 120 140 160 0 0.5 1 1.5 2 Time[min] D ru g In fu si o n R a te [m g m in − 1] Propofol infusion Remifentanil infusion
Figure 3.13: Evolution of the DoA, of Patient 9, and of the dosages of propofol and of remifentanil, assuming changes in the reference profiles (z∗ = 50 from the beginning
till t = 90 min, z∗ = 40 from t = 90 min till t = 120 min, and z∗ = 50 from then on)
and assuming changes in the value of the drugs proportions (ρ = 0.2 in the first forty minutes and from then on it was changed to ρ = 3). λ = 1.
3.2.4
A Strategy to Identify the Parameters of the PPM
A simple technique to identify the parameters of the PPM is presented in this subsec-tion, based on the patient’s response to the administration of fixed drug doses. This technique is made possible due to the simple structure of this parameter parsimonious model. More concretely, in order to identify the parameters γ, η, µ, α, a constant dose
(step) of remifentanil is administered as a single drug during the first t1 minutes, after
which a constant dose (step) of propofol is cumulatively delivered during the next t2
minutes. This is in accordance with what happens in many surgeries, where, to avoid patient discomfort, the administration of the analgesic remifentanil precedes the one of the hypnotic propofol.
The step response for the effect concentration of remifentanil, cr
step(t), by administering
remifentanil, is obtained by the inverse Laplace transform
crstep(t) = L−1 Hr(s)1 s = 3e−2ηt−3e−ηt− e−3ηt+ 1. (3.36)
Equivalentely, the step response for the effect concentration of propofol, cp
step(t), by
administering propofol, is obtained by the inverse Laplace transform
cpstep(t) = L−1 Hp(s)1 s = 54e−9αt− 5 4e −αt− e−10αt+ 1. (3.37)
Thus, when the step of remifentanil, ur
step, is administered, note that in this case
cp
e = 0 and hence Up = 0, the BIS response of the PPM, z(t) = zrstep(t), is given by
